# Properties

 Label 80.10.a.h Level $80$ Weight $10$ Character orbit 80.a Self dual yes Analytic conductor $41.203$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 80.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$41.2028668931$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6049})$$ Defining polynomial: $$x^{2} - x - 1512$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 40) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{6049}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 46 - \beta ) q^{3} + 625 q^{5} + ( 3454 + 37 \beta ) q^{7} + ( 6629 - 92 \beta ) q^{9} +O(q^{10})$$ $$q + ( 46 - \beta ) q^{3} + 625 q^{5} + ( 3454 + 37 \beta ) q^{7} + ( 6629 - 92 \beta ) q^{9} + ( 4040 + 166 \beta ) q^{11} + ( 55974 - 252 \beta ) q^{13} + ( 28750 - 625 \beta ) q^{15} + ( -163766 - 956 \beta ) q^{17} + ( 578340 - 3484 \beta ) q^{19} + ( -736368 - 1752 \beta ) q^{21} + ( 528626 - 9517 \beta ) q^{23} + 390625 q^{25} + ( 1625548 + 8822 \beta ) q^{27} + ( -2106130 - 5912 \beta ) q^{29} + ( 5680564 + 24134 \beta ) q^{31} + ( -3830696 + 3596 \beta ) q^{33} + ( 2158750 + 23125 \beta ) q^{35} + ( -3625930 - 51896 \beta ) q^{37} + ( 8672196 - 67566 \beta ) q^{39} + ( 6515218 + 182956 \beta ) q^{41} + ( 23967038 - 8701 \beta ) q^{43} + ( 4143125 - 57500 \beta ) q^{45} + ( -15457146 + 135245 \beta ) q^{47} + ( 4700833 + 255596 \beta ) q^{49} + ( 15598140 + 119790 \beta ) q^{51} + ( 50461054 + 32140 \beta ) q^{53} + ( 2525000 + 103750 \beta ) q^{55} + ( 110902504 - 738604 \beta ) q^{57} + ( 23681268 - 699384 \beta ) q^{59} + ( 101817214 - 342256 \beta ) q^{61} + ( -59466618 - 72495 \beta ) q^{63} + ( 34983750 - 157500 \beta ) q^{65} + ( 29436426 + 145537 \beta ) q^{67} + ( 254590128 - 966408 \beta ) q^{69} + ( 174950396 - 387506 \beta ) q^{71} + ( 35580194 + 2649812 \beta ) q^{73} + ( 17968750 - 390625 \beta ) q^{75} + ( 162565992 + 722844 \beta ) q^{77} + ( 226043672 - 862932 \beta ) q^{79} + ( -269160511 + 591100 \beta ) q^{81} + ( 122432718 + 2988231 \beta ) q^{83} + ( -102353750 - 597500 \beta ) q^{85} + ( 46164772 + 1834178 \beta ) q^{87} + ( -128036630 - 2272296 \beta ) q^{89} + ( -32269308 + 1200630 \beta ) q^{91} + ( -322640320 - 4570400 \beta ) q^{93} + ( 361462500 - 2177500 \beta ) q^{95} + ( -92475286 + 9414300 \beta ) q^{97} + ( -342740152 + 728734 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 92q^{3} + 1250q^{5} + 6908q^{7} + 13258q^{9} + O(q^{10})$$ $$2q + 92q^{3} + 1250q^{5} + 6908q^{7} + 13258q^{9} + 8080q^{11} + 111948q^{13} + 57500q^{15} - 327532q^{17} + 1156680q^{19} - 1472736q^{21} + 1057252q^{23} + 781250q^{25} + 3251096q^{27} - 4212260q^{29} + 11361128q^{31} - 7661392q^{33} + 4317500q^{35} - 7251860q^{37} + 17344392q^{39} + 13030436q^{41} + 47934076q^{43} + 8286250q^{45} - 30914292q^{47} + 9401666q^{49} + 31196280q^{51} + 100922108q^{53} + 5050000q^{55} + 221805008q^{57} + 47362536q^{59} + 203634428q^{61} - 118933236q^{63} + 69967500q^{65} + 58872852q^{67} + 509180256q^{69} + 349900792q^{71} + 71160388q^{73} + 35937500q^{75} + 325131984q^{77} + 452087344q^{79} - 538321022q^{81} + 244865436q^{83} - 204707500q^{85} + 92329544q^{87} - 256073260q^{89} - 64538616q^{91} - 645280640q^{93} + 722925000q^{95} - 184950572q^{97} - 685480304q^{99} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 39.3877 −38.3877
0 −109.551 0 625.000 0 9209.37 0 −7681.66 0
1.2 0 201.551 0 625.000 0 −2301.37 0 20939.7 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.10.a.h 2
4.b odd 2 1 40.10.a.b 2
5.b even 2 1 400.10.a.o 2
5.c odd 4 2 400.10.c.o 4
8.b even 2 1 320.10.a.o 2
8.d odd 2 1 320.10.a.p 2
12.b even 2 1 360.10.a.a 2
20.d odd 2 1 200.10.a.d 2
20.e even 4 2 200.10.c.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.10.a.b 2 4.b odd 2 1
80.10.a.h 2 1.a even 1 1 trivial
200.10.a.d 2 20.d odd 2 1
200.10.c.e 4 20.e even 4 2
320.10.a.o 2 8.b even 2 1
320.10.a.p 2 8.d odd 2 1
360.10.a.a 2 12.b even 2 1
400.10.a.o 2 5.b even 2 1
400.10.c.o 4 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 92 T_{3} - 22080$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(80))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 92 T + 17286 T^{2} - 1810836 T^{3} + 387420489 T^{4}$$
$5$ $$( 1 - 625 T )^{2}$$
$7$ $$1 - 6908 T + 59513006 T^{2} - 278762717156 T^{3} + 1628413597910449 T^{4}$$
$11$ $$1 - 8080 T + 4065472006 T^{2} - 19052217343280 T^{3} + 5559917313492231481 T^{4}$$
$13$ $$1 - 111948 T + 22805544638 T^{2} - 1187152495808604 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$
$17$ $$1 + 327532 T + 241881460294 T^{2} + 38841324364815404 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$
$19$ $$1 - 1156680 T + 686155308982 T^{2} - 373246406267013720 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$
$23$ $$1 - 1057252 T + 1690239470158 T^{2} - 1904272253637079676 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$
$29$ $$1 + 4212260 T + 32604383130814 T^{2} + 61107870708313953940 T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$31$ $$1 - 11361128 T + 71055092544062 T^{2} -$$$$30\!\cdots\!88$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$
$37$ $$1 + 7251860 T + 207906306187118 T^{2} +$$$$94\!\cdots\!20$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$
$41$ $$1 - 13030436 T - 112698304084010 T^{2} -$$$$42\!\cdots\!96$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$
$43$ $$1 - 47934076 T + 1577772318092534 T^{2} -$$$$24\!\cdots\!68$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$
$47$ $$1 + 30914292 T + 2034610190905950 T^{2} +$$$$34\!\cdots\!64$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$
$53$ $$1 - 100922108 T + 9120851179993582 T^{2} -$$$$33\!\cdots\!64$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$
$59$ $$1 - 47362536 T + 6051611540480326 T^{2} -$$$$41\!\cdots\!04$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$
$61$ $$1 - 203634428 T + 30920737906297022 T^{2} -$$$$23\!\cdots\!48$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$
$67$ $$1 - 58872852 T + 54767076047787046 T^{2} -$$$$16\!\cdots\!44$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$
$71$ $$1 - 349900792 T + 118671349360183822 T^{2} -$$$$16\!\cdots\!52$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$73$ $$1 - 71160388 T - 50883178339169962 T^{2} -$$$$41\!\cdots\!44$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$
$79$ $$1 - 452087344 T + 272781342616725918 T^{2} -$$$$54\!\cdots\!36$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$
$83$ $$1 - 244865436 T + 172811505943449574 T^{2} -$$$$45\!\cdots\!08$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$
$89$ $$1 + 256073260 T + 592174274852066582 T^{2} +$$$$89\!\cdots\!40$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$
$97$ $$1 + 184950572 T - 615454564650127770 T^{2} +$$$$14\!\cdots\!24$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$